Error bounds for kernel-based numerical differentiation
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The literature on meshless methods shows that kernel-based numerical differentiation formulae are robust and provide high accuracy at low cost. This paper analyzes the error of such formulas, using the new technique of growth functions. It allows to bypass certain technical assumptions that were needed to prove the standard error bounds on interpolants and their derivatives. Since differentiation formulas based on polynomials also have error bounds in terms of growth functions, we have a convenient way to compare kernel-based and polynomial-based formulas. It follows that kernel-based formulas are comparable in accuracy to the best possible polynomial-based formulas. A variety of examples is provided.
The authors are grateful to the Alexander von Humboldt Foundation for the financial support that helped them to get together and work on this project for several days in September 2011.
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